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UPSC NDA I 2016 Mathematics Solved Paper

Question Numbers: 61-64

Let

f: \mathbb{R} \to \mathbb{R}

be a function such that

f(x) = x^{3} + x^{2} f'(1) + x f''(2) + f'''(3)

for

x \in \mathbb{R}

Question : 63 of 120

Marks: +1, -0

What is f’’’(10) equal to?

Solution:

Concept:

Differentiation using the power rule and treating constants like

f'(1)

f''(2)

f'''(3)

as fixed numbers.

Explanation:

Given

f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3)

Differentiate with respect to

x

f'(x) = 3x^2 + 2x f'(1) + f''(2)

Differentiate again:

f''(x) = 6x + 2 f'(1)

Differentiate a third time:

f'''(x) = 6

Since

f'''(x)

is constant

6

for all

x

, substituting

x=10

gives

f'''(10)=6

Answer:

6

(Option C).

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